On a conjecture of Beukers and some related congruences (Q1803364)

Consider the numbers \(u(n)=\sum_{k=0}^ n {n\choose k} {{n+k} \choose k}\). Let \(p\) be a prime of the form \(p=4f+1\). Write \(p=a^ 2+b^ 2\) with \(a\equiv 1\pmod 4\). The author proves with elementary means that \(u(2f)\equiv(-1)^ f (2a-p/2a)\pmod {p^ 2}\), thus proving a conjecture once formulated by the reviewer. It must be remarked that this congruence was proved earlier by \textit{L. Van Hamme} [Proc. Conf. on \(p\)-adic analysis, Houthalen/Belgium 1986, 189-195 (1986; Zbl 0634.10004)] and a generalization by \textit{M. Coster} and \textit{L. Van Hamme} [J. Number Theory 38, 265-286 (1991; Zbl 0732.11021)]. Several other congruences, such as \(u(p-1)\equiv 3-2^ p \pmod {p^ 2}\), \(u(p)\equiv 1+2^ p \pmod{p^ 2}\) for any prime \(p\), are also proved.